Problem: $ F = \left[\begin{array}{rrr}-1 & 4 & 1 \\ -1 & 5 & 5\end{array}\right]$ $ B = \left[\begin{array}{rr}-2 & 3 \\ -2 & 0 \\ -2 & -1\end{array}\right]$ What is $ F B$ ?
Because $ F$ has dimensions $(2\times3)$ and $ B$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ F B = \left[\begin{array}{rrr}{-1} & {4} & {1} \\ {-1} & {5} & {5}\end{array}\right] \left[\begin{array}{rr}{-2} & \color{#DF0030}{3} \\ {-2} & \color{#DF0030}{0} \\ {-2} & \color{#DF0030}{-1}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ F$ , with the corresponding elements in column $j$ of the second matrix, $ B$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ F$ with the first element in ${\text{column }1}$ of $ B$ , then multiply the second element in ${\text{row }1}$ of $ F$ with the second element in ${\text{column }1}$ of $ B$ , and so on. Add the products together. $ \left[\begin{array}{rr}{-1}\cdot{-2}+{4}\cdot{-2}+{1}\cdot{-2} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ F$ with the corresponding elements in ${\text{column }1}$ of $ B$ and add the products together. $ \left[\begin{array}{rr}{-1}\cdot{-2}+{4}\cdot{-2}+{1}\cdot{-2} & ? \\ {-1}\cdot{-2}+{5}\cdot{-2}+{5}\cdot{-2} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ F$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ B$ and add the products together. $ \left[\begin{array}{rr}{-1}\cdot{-2}+{4}\cdot{-2}+{1}\cdot{-2} & {-1}\cdot\color{#DF0030}{3}+{4}\cdot\color{#DF0030}{0}+{1}\cdot\color{#DF0030}{-1} \\ {-1}\cdot{-2}+{5}\cdot{-2}+{5}\cdot{-2} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{-1}\cdot{-2}+{4}\cdot{-2}+{1}\cdot{-2} & {-1}\cdot\color{#DF0030}{3}+{4}\cdot\color{#DF0030}{0}+{1}\cdot\color{#DF0030}{-1} \\ {-1}\cdot{-2}+{5}\cdot{-2}+{5}\cdot{-2} & {-1}\cdot\color{#DF0030}{3}+{5}\cdot\color{#DF0030}{0}+{5}\cdot\color{#DF0030}{-1}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}-8 & -4 \\ -18 & -8\end{array}\right] $